Embedded newform 1728.2.a.d.1.1

• Label: 1728.2.a.d.1.1
• Level: $1728$
• Weight: $2$
• Character: 1728.1
• Self dual: yes
• Analytic conductor: $13.798$
• Analytic rank: $1$
• Dimension: $1$
• CM: no
• Inner twists: $1$

Newspace parameters

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.a (trivial)

Newform invariants

Self dual:	yes
Analytic conductor:	\(13.7981494693\)
Analytic rank:	\(1\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	no (minimal twist has level 54)
Fricke sign:	\(+1\)
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	\(\chi\)	\(=\)	1728.1

$q$-expansion

\(f(q)\)

\(=\)

\(q-3.00000 q^{5} +1.00000 q^{7} -3.00000 q^{11} +4.00000 q^{13} +2.00000 q^{19} +6.00000 q^{23} +4.00000 q^{25} -6.00000 q^{29} -5.00000 q^{31} -3.00000 q^{35} -2.00000 q^{37} -6.00000 q^{41} -10.0000 q^{43} -6.00000 q^{47} -6.00000 q^{49} -9.00000 q^{53} +9.00000 q^{55} +12.0000 q^{59} -8.00000 q^{61} -12.0000 q^{65} +14.0000 q^{67} -7.00000 q^{73} -3.00000 q^{77} -8.00000 q^{79} -3.00000 q^{83} -18.0000 q^{89} +4.00000 q^{91} -6.00000 q^{95} -1.00000 q^{97} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	0	0
			\(3\)	0	0
\(5\)	−3.00000	−1.34164				−0.670820	−	0.741620i	\(-0.734058\pi\)
			−0.670820	+	0.741620i	\(0.734058\pi\)
\(7\)	1.00000	0.377964	0.188982	−	0.981981i	\(-0.439481\pi\)
			0.188982	+	0.981981i	\(0.439481\pi\)
\(11\)	−3.00000	−0.904534	−0.452267	−	0.891883i	\(-0.649385\pi\)
			−0.452267	+	0.891883i	\(0.649385\pi\)
\(13\)	4.00000	1.10940	0.554700	−	0.832050i	\(-0.312833\pi\)
			0.554700	+	0.832050i	\(0.312833\pi\)
\(17\)	0	0		−	1.00000i	\(-0.5\pi\)
					1.00000i	\(0.5\pi\)
\(19\)	2.00000	0.458831	0.229416	−	0.973329i	\(-0.426318\pi\)
			0.229416	+	0.973329i	\(0.426318\pi\)
\(23\)	6.00000	1.25109	0.625543	−	0.780189i	\(-0.284877\pi\)
			0.625543	+	0.780189i	\(0.284877\pi\)
\(29\)	−6.00000	−1.11417	−0.557086	−	0.830455i	\(-0.688081\pi\)
			−0.557086	+	0.830455i	\(0.688081\pi\)
\(31\)	−5.00000	−0.898027	−0.449013	−	0.893525i	\(-0.648224\pi\)
			−0.449013	+	0.893525i	\(0.648224\pi\)
\(37\)	−2.00000	−0.328798	−0.164399	−	0.986394i	\(-0.552568\pi\)
			−0.164399	+	0.986394i	\(0.552568\pi\)
\(41\)	−6.00000	−0.937043	−0.468521	−	0.883452i	\(-0.655213\pi\)
			−0.468521	+	0.883452i	\(0.655213\pi\)
\(43\)	−10.0000	−1.52499	−0.762493	−	0.646997i	\(-0.776025\pi\)
			−0.762493	+	0.646997i	\(0.776025\pi\)
\(47\)	−6.00000	−0.875190	−0.437595	−	0.899172i	\(-0.644170\pi\)
			−0.437595	+	0.899172i	\(0.644170\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.a.d.1.1		1
3.2	odd	2		1728.2.a.z.1.1		1
4.3	odd	2		1728.2.a.c.1.1		1
8.3	odd	2		54.2.a.a.1.1	✓	1
8.5	even	2		432.2.a.g.1.1		1
12.11	even	2		1728.2.a.y.1.1		1
24.5	odd	2		432.2.a.b.1.1		1
24.11	even	2		54.2.a.b.1.1	yes	1
40.3	even	4		1350.2.c.b.649.2		2
40.19	odd	2		1350.2.a.r.1.1		1
40.27	even	4		1350.2.c.b.649.1		2
56.27	even	2		2646.2.a.a.1.1		1
72.5	odd	6		1296.2.i.o.865.1		2
72.11	even	6		162.2.c.b.109.1		2
72.13	even	6		1296.2.i.c.865.1		2
72.29	odd	6		1296.2.i.o.433.1		2
72.43	odd	6		162.2.c.c.109.1		2
72.59	even	6		162.2.c.b.55.1		2
72.61	even	6		1296.2.i.c.433.1		2
72.67	odd	6		162.2.c.c.55.1		2
88.43	even	2		6534.2.a.bc.1.1		1
104.51	odd	2		9126.2.a.u.1.1		1
120.59	even	2		1350.2.a.h.1.1		1
120.83	odd	4		1350.2.c.k.649.1		2
120.107	odd	4		1350.2.c.k.649.2		2
168.83	odd	2		2646.2.a.bd.1.1		1
264.131	odd	2		6534.2.a.b.1.1		1
312.155	even	2		9126.2.a.r.1.1		1

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.2.a.a.1.1	✓	1	8.3	odd	2
54.2.a.b.1.1	yes	1	24.11	even	2
162.2.c.b.55.1		2	72.59	even	6
162.2.c.b.109.1		2	72.11	even	6
162.2.c.c.55.1		2	72.67	odd	6
162.2.c.c.109.1		2	72.43	odd	6
432.2.a.b.1.1		1	24.5	odd	2
432.2.a.g.1.1		1	8.5	even	2
1296.2.i.c.433.1		2	72.61	even	6
1296.2.i.c.865.1		2	72.13	even	6
1296.2.i.o.433.1		2	72.29	odd	6
1296.2.i.o.865.1		2	72.5	odd	6
1350.2.a.h.1.1		1	120.59	even	2
1350.2.a.r.1.1		1	40.19	odd	2
1350.2.c.b.649.1		2	40.27	even	4
1350.2.c.b.649.2		2	40.3	even	4
1350.2.c.k.649.1		2	120.83	odd	4
1350.2.c.k.649.2		2	120.107	odd	4
1728.2.a.c.1.1		1	4.3	odd	2
1728.2.a.d.1.1		1	1.1	even	1	trivial
1728.2.a.y.1.1		1	12.11	even	2
1728.2.a.z.1.1		1	3.2	odd	2
2646.2.a.a.1.1		1	56.27	even	2
2646.2.a.bd.1.1		1	168.83	odd	2
6534.2.a.b.1.1		1	264.131	odd	2
6534.2.a.bc.1.1		1	88.43	even	2
9126.2.a.r.1.1		1	312.155	even	2
9126.2.a.u.1.1		1	104.51	odd	2